Bottom Line:
By exploiting the strong non-linear couplings between quantized excitations emerging when superconducting qubits are coupled, we show how to engineer gauge invariant Hamiltonians, including ring-exchange and four-body Ising interactions.We demonstrate that, despite decoherence and disorder effects, minimal circuit instances allow us to investigate properties such as the dynamics of electric flux strings, signaling confinement in gauge invariant field theories.The experimental realization of these models in larger superconducting circuits could address open questions beyond current computational capability.

A quantum simulator of [Formula: see text] lattice gauge theories can be implemented with superconducting circuits. This allows the investigation of confined and deconfined phases in quantum link models, and of valence bond solid and spin liquid phases in quantum dimer models. Fractionalized confining strings and the real-time dynamics of quantum phase transitions are accessible as well. Here we show how state-of-the-art superconducting technology allows us to simulate these phenomena in relatively small circuit lattices. By exploiting the strong non-linear couplings between quantized excitations emerging when superconducting qubits are coupled, we show how to engineer gauge invariant Hamiltonians, including ring-exchange and four-body Ising interactions. We demonstrate that, despite decoherence and disorder effects, minimal circuit instances allow us to investigate properties such as the dynamics of electric flux strings, signaling confinement in gauge invariant field theories. The experimental realization of these models in larger superconducting circuits could address open questions beyond current computational capability.

f000025: Circuit lattice for the simulation of the model (11). Every plaquette contains a qubit (e.g. a transmon) on the links. Hopping and Kerr interactions of local excitations are enabled by a capacitor in parallel with a Josephson junction connecting neighboring qubits, giving rise to interactions, and–perturbatively–to ring-exchange dynamics. The tunneling term through each vertex may be suppressed by choosing appropriately the value of the parallel capacitor to the Josephson junction.

Mentions:
Let us consider the general circuit lattice depicted in Fig. 5. On each link the lowest two energy levels of a strongly coupled superconducting circuit (qubit) are used to implement an effective spin system, representing the gauge field, as described in Section 2. Neighboring spins on each plaquette and across each node are connected by Josephson junctions, which induce nearest-neighbor interactions. By an appropriate choice of parameters, the resulting Hamiltonian of the circuit lattice takes the form (13) where is the bare frequency splitting between qubit states (the sum involves nearest-neighbor lattice sites). The interactions and are diagonal coupling constants for qubits located on opposite sides of each lattice site and neighboring qubits within the same plaquette, respectively [see Fig. 5] (the sum denotes qubits around vertices, and the sum involves nearest-neighbor links around a plaquette). In addition, neighboring qubits located within the same plaquette are coupled by a small hopping term . By defining and omitting an overall frequency shift, we can rewrite the Hamiltonian (13) as (14) where for each site is the gauge generator introduced above. Under the assumption that the system is initially prepared in the subspace of states with exactly two spins up and two spins down around each site, for all , transitions out of this subspace are suppressed by a large energy gap . In the limit we can use perturbation theory to derive an effective Hamiltonian for this subspace, which is given by(15) where (16)J=4μ2Ω,V=V′−J. Apart from the overall qubit energy , which does not affect the dynamics in the gauge invariant subspace, this effective Hamiltonian reproduces the gauge invariant model (11). In particular, taking , the standard ring-exchange interaction (10) is recovered. An interaction of the type (arising in the RK model) requires an additional circuit element, which will be discussed in Section 3.4.

f000025: Circuit lattice for the simulation of the model (11). Every plaquette contains a qubit (e.g. a transmon) on the links. Hopping and Kerr interactions of local excitations are enabled by a capacitor in parallel with a Josephson junction connecting neighboring qubits, giving rise to interactions, and–perturbatively–to ring-exchange dynamics. The tunneling term through each vertex may be suppressed by choosing appropriately the value of the parallel capacitor to the Josephson junction.

Mentions:
Let us consider the general circuit lattice depicted in Fig. 5. On each link the lowest two energy levels of a strongly coupled superconducting circuit (qubit) are used to implement an effective spin system, representing the gauge field, as described in Section 2. Neighboring spins on each plaquette and across each node are connected by Josephson junctions, which induce nearest-neighbor interactions. By an appropriate choice of parameters, the resulting Hamiltonian of the circuit lattice takes the form (13) where is the bare frequency splitting between qubit states (the sum involves nearest-neighbor lattice sites). The interactions and are diagonal coupling constants for qubits located on opposite sides of each lattice site and neighboring qubits within the same plaquette, respectively [see Fig. 5] (the sum denotes qubits around vertices, and the sum involves nearest-neighbor links around a plaquette). In addition, neighboring qubits located within the same plaquette are coupled by a small hopping term . By defining and omitting an overall frequency shift, we can rewrite the Hamiltonian (13) as (14) where for each site is the gauge generator introduced above. Under the assumption that the system is initially prepared in the subspace of states with exactly two spins up and two spins down around each site, for all , transitions out of this subspace are suppressed by a large energy gap . In the limit we can use perturbation theory to derive an effective Hamiltonian for this subspace, which is given by(15) where (16)J=4μ2Ω,V=V′−J. Apart from the overall qubit energy , which does not affect the dynamics in the gauge invariant subspace, this effective Hamiltonian reproduces the gauge invariant model (11). In particular, taking , the standard ring-exchange interaction (10) is recovered. An interaction of the type (arising in the RK model) requires an additional circuit element, which will be discussed in Section 3.4.

Bottom Line:
By exploiting the strong non-linear couplings between quantized excitations emerging when superconducting qubits are coupled, we show how to engineer gauge invariant Hamiltonians, including ring-exchange and four-body Ising interactions.We demonstrate that, despite decoherence and disorder effects, minimal circuit instances allow us to investigate properties such as the dynamics of electric flux strings, signaling confinement in gauge invariant field theories.The experimental realization of these models in larger superconducting circuits could address open questions beyond current computational capability.

A quantum simulator of [Formula: see text] lattice gauge theories can be implemented with superconducting circuits. This allows the investigation of confined and deconfined phases in quantum link models, and of valence bond solid and spin liquid phases in quantum dimer models. Fractionalized confining strings and the real-time dynamics of quantum phase transitions are accessible as well. Here we show how state-of-the-art superconducting technology allows us to simulate these phenomena in relatively small circuit lattices. By exploiting the strong non-linear couplings between quantized excitations emerging when superconducting qubits are coupled, we show how to engineer gauge invariant Hamiltonians, including ring-exchange and four-body Ising interactions. We demonstrate that, despite decoherence and disorder effects, minimal circuit instances allow us to investigate properties such as the dynamics of electric flux strings, signaling confinement in gauge invariant field theories. The experimental realization of these models in larger superconducting circuits could address open questions beyond current computational capability.